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Introduction to Atomic Physics Lab Report Jeppe Søgaard Juul Esben Tore Mølgaard June 1, 2007 1 1.1 Theory Atomic structure and electron energies Atoms are build up by a positive core and negatively charged electrons circulating around it. The wave functions of the electrons are distinguished by their quantum numbers, which in the Bohr model for hydrogen are given by the principal number n, the orbital angular momentum l and its z-component ml , the spin s and its z-component ms . In fine structure theory relativistic effects are taken into account. That is a relativistic correction to the kinetic energy, a coupling between spin angular momentum and orbital angular momentum, and the Darwin term. As the hamiltonian of the spin-orbit coupling does not commute with the operators Lz and Sz , ml and ms are no longer good quantum numbers. Instead we use the total angular momentum j and its z-component mj . Depending on the quantum numbers, the electrons have different energies. According to the Bohr model for hydrogen, the energies only depend on the principal quantum number n: The energy increases for larger n. The fine structure removes some of this degeneracy: According to this, the energy also depends on the the total angular momentum j. Again the energy increases with the total angular momentum. If the spin of the nucleus is taken into account, the energy becomes dependent on this as well, giving rise to the hyperfine structure. Quantum electro dynamics also gives rise to a shift in energy depended of the quantum number l. These effects are not relevant in this exercise. 1.2 Many electron atoms and transitions Other atoms than hydrogen has many electrons circulating around the nucleus which interact with each other. The inner electrons (with low n and l values) are ’screening’ the positive nucleus from the outer electrons, which are thereby looser bound to the atom. The higher n and l the electrons have, the farther away the electron is from the nucleus and the larger will this screening effect be. The energy of the electrons will thus increase for higher l. The a Grotrian digram for sodium can be seen in figure 2. It is possible for an electron to make a transition between two quantum states through emission of a photon. As the photon has a spin of 1, there must be a difference in orbital angular momentum of 1 between the two states: ∆l = ±1. The emitted photons have 1 a well defined wavelengths proportional to the energy difference of the quantum states, this gives rise to spectral lines characteristic of the different elements. As indicated in figure 2 this energy difference between the states is dominated by the difference in quantum numbers n and l, due to Bohr’s atom model and the screening effects. Fine structure will, though, cause a slight energy difference depending the the quantum number j. This will be seen as splitting of the spectral lines. 2 Setup and Procedure In the experiment we study the spectral lines of sodium. Different wavelengths of sodium light are separated in a crystal in a spectrometer, and the spectral lines can hereby be observed and the wavelengths measured. In part a of the exercise we use a low pressure Na-lamp to calibrate the spectrometer. We measured the wavelength of the characteristic yellow double line 10 times going from low to high frequencies and 10 times going from high to low frequencies, and set the spectrometer so the mean value corresponded to the tabular values. Afterward we studied the full spectrum, noting the wavelength, colour, sharpness and intensity of the all the lines we could identify, and the number of splittings. In part b of the exercise we studied the spectrum from a high pressure Na lamp, noting the same parameters as in part a. 3 3.1 Results and Discussion Part a We measured the center of the yellow doublet, which should be at λ = 5893 Å, to be From low frequency From high frequency Mean (Å) 5890 5885 σ (Å) 2.29 2.07 We measured the following wavelengths and identified them hλi (Å) σ (Å) λtabular (Å) 6164 5 6157 5898 4 5892 5692 3 5685 5155 3 5151 4990 3 4980 4757 3 4749 4674 3 4666 4556 4 4543 4502 3 4496 Tabular line splitting Sharp red doublet Sharp yellow doublet Sharp green doublet Sharp cyan doublet Sharp blue doublet Diffuse violet singlet Diffuse violet doublet Diffuse violet singlet Diffuse violet singlet Desription doublet doublet triplet doublet doublet doublet doublet doublet doublet Transition 3p-5s 3s-3p 3p-4d 3p-6s 3p-5d 3p-7s 3p-6d 3p-8s 3p-7d The reason we identify the green line at λ = 5685 Å as a doublet is that two of the lines lie so close that they are almost impossible to distinguish. The reason we identify the violet lines as singlets is that they are so faint that it is impossible to distinguish. 2 We do not see the expected connection between the spectroscopic designation (s,p,d,. . . ) and the sharpness of the spectral lines. This is most likely because it was very hard to see, and we mostly designated sharp and diffuse based on the intensity. We made a plot of the tabular values of the wavelength and our measured wavelength and got the calibration curve shown in figure 1. Figure 1: Calibration curve for our experiment. A Grotrian diagram of the transitions can be seen as figure 2. 3.2 Part b When we measured the spectral lines from the high preassure lamp, we found several of the same lines as using the low preassure lamp, but also five lines that did not belong in the sodium spectrum. Using available litterature, and in particular the supplied Bright Line Spectra figure, we determined that the lines that were not sodium were most likely argon. We reached this conclusion because argon has lines corresponding to all five lines we could not identify as being from sodium, we got numerical values for these lines from the CRC Handbook of Chemistry and Physics 84th edition, however there was no data for the line splitting. Using the calibration curve (figure 1), we got the following calibrated wavelengths, which we compare to the tabluar values of sodium and argon. 3 Figure 2: Grotrian diagram of the visible spectrum of sodium. λcalibrated (Å) λN a (Å) λA (Å) 6745 6753 6193 6157 5926 5892 5713 5713 5485 5470 5361 5346 5164 5151 4995 4980 4370 4364 Desription Sharp red singlet Sharp red doublet Black singlet Diffuse yellow singlet Sharp green/cyan doublet Sharp black doublet Green singlet Diffuse green singlet Diffuse violet singlet Tabular line splitting doublet doublet doublet doublet It is interesting to note that the yellow line we for the low preassure lamp had around λ = 5893 Å and the cyan line around λ = 5151 Å have turned black. This is because the high pressure send out light in the full visible spectrum, and the black lines come from 4 absorbtion by the sodium atoms, whereas the coloured lines came from emission. 4 Conclusion Using a spectrometer we studied the spectral lines of light emitted from low and high pressure Na lamps. We found a clear correspondence between the the wavelengths of the observed lines and the tabular values. In the high pressure Na lamp we found five lines originating from a Argon. We could not find a clear connection between the sharpness of the lines and the quantum number l, and it was not always possible to identify the correct number of splittings of the spectral lines, caused by fine structure theory. 5